Words: 12,268 (Appendix/SI: 2,373)

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*corresponding author: rhcr@unimelb.edu.au

Hybrid life cycle inventory methods – a review

Robert H. Crawforda*, Paul-Antoine Bontincka, André Stephana, Thomas Wiedmannb, Man

Yub

a Faculty of Architecture, Building and Planning, The University of Melbourne, Parkville,

Victoria 3010, Australia

b School of Civil and Environmental Engineering, UNSW Sydney, NSW 2052, Australia

Abstract

A variety of methods can be used to compile a life cycle inventory (LCI) as part of a

life cycle assessment (LCA) study. Hybrid LCI methods attempt to address the limitations

inherent in more traditional process and input-output (IO) LCI methods. This paper provides

an overview of the different hybrid LCI methods currently in use in an attempt to provide

greater clarity around how each method is applied and their specific strengths and

weaknesses. A search of publications quoting the use of hybrid LCI was undertaken for the

period from 2010 to 2015, identifying 97 peer-reviewed publications referencing the use of

a hybrid LCI. In over one third of the literature analysed, authors only refer to their analysis

as a hybrid LCI, without naming the actual method used, making it difficult to fully

understand the method used and any potential limitations. Based on the way in which the

various hybrid methods are applied and their existing use, the authors propose a set of clear

definitions for existing hybrid LCI methods. This assists in creating a better understanding of,

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and confidence in applying hybrid LCI methods amongst LCA practitioners, potentially

leading to a greater uptake of hybrid LCI.

Keywords

Life cycle assessment; life cycle inventory analysis; input-output analysis; process analysis;

hybrid analysis.

1. Introduction

Assessing and minimising the effects of human activities on the environment is part

of the long-term goal to achieve sustainable development within the limits of Earth's

resources and the capacity for natural systems to absorb disruptions from anthropogenic

activity without collapsing (Brundtland et al., 1987; IPCC, 2013, 2014; Meadows et al., 1972).

In many countries, products, services and processes or, on a larger scale, construction and

infrastructure projects are routinely assessed over a variety of environmental indicators.

The goal of these assessments is to identify current environmental performance, areas for

improvement and the extent of benefits achieved from improvement activities. The results

of these assessments are also often used in environmental decision-making, such as

choosing a specific product or design over another, or informing and implementing policies

for sustainability (Hellweg and Canals, 2014). It is therefore evident that the quality,

reliability and robustness of these assessments are critical in order to inform effective

decisions and select appropriate solutions for improving environmental performance.

The development of assessment techniques for the estimation of the environmental

effects of a product system has been an area of research for at least the past fifty years. It

has borne a variety of methods, with the most recent of these being life cycle assessment

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(LCA) (Klöpffer, 2014). As described in the ISO 14040 standard, LCA is “a compilation and

evaluation of the inputs, outputs and the potential environmental impacts of a product

system throughout its life cycle” (ISO, 2006).

An integral part of an LCA is the production of a life cycle inventory, listing the inputs

and outputs associated with the product system under study. Three broad approaches for

compiling a life cycle inventory (LCI) are commonly used. The first is known as process

analysis, or a bottom-up approach, where the product system analysed is broken down into

a series of processes representing the life cycle of a product. The second is environmentally-

extended input-output analysis (EEIOA), or a top-down approach, which is rooted in

macroeconomics. The last is known as hybrid analysis, and involves combining the first two

approaches. Each of these approaches requires modelling a system, using either specific

production processes or entire economic systems.

The development of hybrid methods has been the focus of research for at least the

last 20 years (Bullard et al., 1978; Joshi, 1999; Lenzen, 2000; Lenzen and Crawford, 2009;

Suh and Huppes, 2005; Suh et al., 2004; Treloar, 1997). These methods combine process and

input-output data in a variety of formats, and all fall under the spectrum of hybridisation

between conventional process and input-output analysis.

While hybrid LCI methods have been used for decades, there is a lack of clear and

consistent terminology to describe each of the available methods. This results in a certain

level of confusion within the current body of work focusing on hybrid methods, leading to a

reduced confidence in the hybrid approach. The suite and complexity of methods available

to practitioners is also likely to be a key factor in preventing hybrid LCI methods from

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becoming a standard approach for environmental assessments, as there is no real

consensus over the preferred method. In parallel, most LCAs currently use a conventional

process analysis for the LCI, with some level of awareness that the assessments are not

complete, but little understanding of the potential extent of this incompleteness. Finally,

there is currently no software allowing comprehensive hybrid analysis with the same level of

simplicity and flexibility as currently exists with process analysis focussed software such as

Simapro or GaBi. This is due to the need to combine two distinctively different types of data,

further exacerbating the slow uptake of hybrid LCI methods.

The aim of this paper is to present a consistent framework for classifying and

defining hybrid LCI methods, discuss their respective advantages and limitations and provide

recommendations to facilitate their uptake by researchers and practitioners. A major aspect

of the classification and definition work involved an analysis of the recent literature

referring to the use of hybrid LCI methods, which also permitted the authors to assess the

homogeneity of application of the different hybrid LCI methods. This analysis is, to the

author’s knowledge, the first attempt at a comprehensive overview of the application of

hybrid LCI methods. It builds upon previous reviews by Islam et al. (2016) and Nakamura

and Nansai (2016), which focused on methodological aspects.

A short historical background to the development of hybrid LCI methods is provided

and a standardised terminology and definition for each of the currently available hybrid LCI

methods is proposed. Each method is described, discussing its theoretical background and

providing some examples of its recent application. Mathematical equations pertaining to

the application of each method are provided in Appendix B. This analysis will be used to

draw examples of inconsistencies in the way each method is applied. The potential reasons

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for the slow uptake of hybrid LCI methods in the LCA community are then discussed, and

specific recommendations aimed at promoting the uptake of these methods are provided.

2. Hybrid analysis in the literature – analysis method

This section outlines the method followed by the authors in searching and analysing

the available literature. This paper aims to analyse a representative sample of recent

publications quoting the use of a hybrid LCI method. A period of five years spanning

between 2010 and 2015 was considered. The temporal boundary was chosen in order to

focus on recent work using these methods, while keeping the scope of the analysis

manageable. It identified 97 peer-reviewed publications.

To guarantee a certain level of quality, only peer-reviewed scientific publications and

theses were selected. In an attempt to produce a representative sample of the literature,

Google Scholar was used in conjunction with the University of Melbourne’s own library

system, and cross-checked with Scopus and Web of Science databases. The keywords listed

in Table 1 were used as initial selection criteria for publications. Different combinations of

these keywords were used to create a listing as representative as possible. It is worth noting

that the analysis rests on the use of online search engines, and thus the representativeness

of the sample used for the analysis is reliant upon not only the list of keywords used by the

authors, but also the efficiency of the search engine used.

Table 1 List of keywords used for the literature search

hybrid IO-LCA integrated hybrid

hybrid LCA/life cycle assessment path exchange method

hybrid LCI/life cycle inventory process-based hybrid

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input-output-based hybrid structural path analysis

input-output hybrid tiered hybrid

The next step was a detailed analysis of each individual paper, and in particular of

the details of the method used by the author/s. Additional information was collected from

each paper, including the location of the authors, the name of the hybrid method as quoted

by the authors and the type of product or process assessed. A detailed list of the

publications analysed can be found in Appendix A.

Table 2 provides a summary of the references to hybrid LCI found in the literature.

These have been grouped according to the four main hybrid LCI methods found; Tiered,

Path Exchange, Matrix Augmentation and Integrated. It shows that Tiered hybrid analysis is

the most commonly used hybrid method, used in 48% of all studies analysed, although there

is considerable inconsistency in the way it is described. On the other hand, a homogeneous

use of the term ‘integrated hybrid’ was observed in the literature. ‘Input-output-based

hybrid’ is a term routinely used by authors describing the PXC and the Matrix Augmentation

methods, which may cause some level of confusion. This is probably due to the fact that

both methods were theorised in the late 1990s, and both were named input-output-based

hybrid by their respective developer. Finally, in over one third of the literature analysed,

authors only refer to their analysis as a hybrid LCA/LCI, without naming the actual method

used, which can be an issue for readers that are unfamiliar with hybrid LCI methods. It can

also make it difficult to fully understand the analysis method used and any potential

limitations.

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Table 2 Summary of the hybrid LCI methods found in the literature (2010-2015)

Method Number of instances

Name of method quoted by the author

Tiered 49 Additive (3); Additive EIO LCA (1); Consumption-weighted LCA (1); Economic balance (1); Economic input-output-based (2); Hybrid input-output (3); Input-output-based (1); MC3 (1); Process-based (3); Stochastic (1); Tiered (12); none (20)

Path Exchange 11 Input-output-based (8); Path Exchange (2); none (1)

Matrix Augmentation

27 Input-output-based (12); Tiered (2); none (13)

Integrated 15 Economic balance (1); Integrated (13); Product equilibrium (1), none (2)

3. Development of hybrid life cycle inventory methods

This section provides a brief overview of the historical development of hybrid life

cycle inventory methods. The intertwined history of input-output, process and hybrid

analysis development started in the 1920s and is still on-going. A selection of key milestones

in this development is shown in Figure 1.

Early in the 20th Century, Wassily Leontief introduced the idea of modelling all

sectors of an economy via a table of inputs and outputs (Leontief, 1928), and later published

the first input-output tables representing the US economy in 1919 and 1929 (Leontief,

1936). In these early publications, he introduced the idea of interdependence between

sectors of the economy, and of representing the economy as a circular system. The

publication of the first input-output tables sparked the creation of a whole new field in

Economic Science, and earned Leontief a Nobel Prize by 1973.

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By the 1960s, interest in assessing the environmental effects of human activities was

growing, due to heightened concerns about the Earth’s limited resources (Goldsmith et al.,

1972; Meadows et al., 1972). Researchers, including Leontief, started experimenting with

using input-output tables to analyse environmental issues (Isard et al., 1968; Leontief,

1970), thus extending the original field of analysis of input-output tables – and creating

environmentally-extended input-output analysis (EEIOA). The first process analyses were

undertaken around the same time, initially focusing on energy and resource analysis (Smith,

1963). As opposed to using macroeconomics, the process analysis approach – then referred

to as Resource and Environmental Profile Analysis (REPA), focused on following the chain of

production in an attempt to take a holistic view of the life cycle of a specific product (Hunt

et al., 1996). These studies originally focused on waste production and energy use and were

increasingly used by the private sector to better understand their production systems,

although interest by public agencies also existed in the 1970s (Hunt et al., 1996). Most

studies were completed confidentially during the late 1970s and all through the 1980s, with

little public interest, until the early 1990s and the development by the Society of

Environmental Toxicology and Chemistry (SETAC) of what is now known as life cycle

assessment (LCA) (Hunt et al., 1996). As research progressed, issues with process and input-

output analysis methods were identified and better understood. In response to this,

researchers began to propose the combination of process and EEIOA in an attempt to

minimise the issues inherent in each of the individual methods.

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Figure 1 Abbreviated timeline of the development of input-output, process and hybrid analysis 2

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3.1. Advantages and limitations of life cycle inventory methods

Process analysis has the advantage of using data specific to the product under study,

enabling the highest possible level of accuracy. On the other hand it suffers from systemic

incompleteness as it is impossible to exhaustively assess the supply chain of any given

product due mainly to cost and time constraints (Crawford, 2008; Lenzen, 2000; Suh et al.,

2004). Three levels of truncation are inherent to a process analysis, including upstream,

downstream and sideways truncation (Crawford, 2011). Upstream truncation relates to the

need to draw a system boundary around the product system being analysed due to time,

budget or data availability constraints. Higher order processes in the supply chain are often

excluded, such as the extraction of a particular raw material at the very start of the supply

chain. Over time, as life cycle inventory databases develop and more is known about supply

chain structures, the number of processes excluded tends to decrease. Downstream

truncation refers to the potential for intermediate manufacturing processes to be omitted –

for instance the manufacturing processes occurring between the production of a material

and the final product. This is typically the case for complex products requiring many

production steps. An example is fabricated products, such as a steel beam, where the

processes associated with the production of the raw materials are included, such as steel,

but not the process of fabricating the beam from raw steel. Finally, sideways truncation

refers to processes that are typically excluded from a process analysis due to either the

assumed insignificance of their contribution to the final results, lack of awareness of their

existence, or a lack of data. This often includes non-material processes such as the provision

of services, or physical inputs considered small enough to be excluded.

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Environmentally-extended input-output analysis enables a systemically complete

coverage of the product system as the data used covers the entire economy. However,

input-output data is aggregated at the economic sector and product category level, which

makes it difficult to assess specific products and differentiate between practices that take

place within the same sector (Lenzen, 2000; Treloar, 1997). EEIOA is also based on the

assumption that monetary flows are a fair indication of the physical flows within an

economy – for instance that it is fair to say that a dollar spent on two different products

from the same sector results in the same resource demands. Also, as input-output tables

only represent the relationships between different sectors of an economy they often

exclude crucial aspects of the life cycle of any given product, such as its use by consumers or

its end-of-life management (Majeau-Bettez et al., 2011). Another crucial aspect often

omitted from EEIOA models is capital goods (Majeau-Bettez et al., 2011; Nakamura and

Nansai, 2016; Williams et al., 2009), which has the potential to significantly affect the results

(Crawford, 2008) These issues have long been recognised and models have been developed

to include capital goods (Lenzen and Treloar, 2004), end-of-life management(Nakamura and

Nansai, 2016), as well as recommendations for modelling the use phase of consumer goods

(Lenzen, 2000) Because of these limitations, EEIOA is most useful for assessing entire

economies or industries, as demonstrated by Achjar et al. (2004), Peters and Hertwich

(2006), and Schuerch et al. (2012).

In an attempt to address the issues inherent in both process analysis and EEIOA,

researchers have proposed the combination of process and input-output data as early as the

1970s (Bullard et al., 1978) into a method known as hybrid analysis. Bullard et al.

proposition led to the development of Tiered hybrid analysis, which is based on a process

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analysis and uses input-output data to fill any remaining higher order gaps in the analysis

and reduce, mainly, upstream truncation errors. It is worth noting here that the raison

d’être of a hybrid method is to provide the most complete and accurate set of information

when compiling the inventory of a product system. As defined in official standards (ISO,

2006), LCA focuses on assessing goods and services, which shifts the macroeconomics focus

of input-output analysis to specific aspects of the economy. Although some methods allow

it, the purpose of a hybrid analysis is often not to provide a complete representation of the

economy using both process and input-output data. Rather, it is to represent the entire

effort required from the economy to create a very specific output.

A number of other hybrid LCI methods have since been proposed, as shown in

Figure 1 (Joshi, 1999; Suh, 2002; Treloar, 1997). These are referred to as Path Exchange

(Treloar, 1997), Matrix Augmentation (Joshi, 1999), and Integrated (Suh, 2002) hybrid LCI

methods. Each of these methods is defined in Section 4, and described in further detail in

Section 5.

4. Defining hybrid life cycle inventory methods

A thorough assessment of the recent literature on hybrid LCI methods shows a lack

of a clear nomenclature to name the methods used to hybridise process and input-output

data, or that the existing nomenclature is not consistently used (see Table 2). It is argued

that there are three main aspects slowing down the uptake of hybrid LCI methods in LCA.

The first is a lack of clarity in the description of the methods used in particular studies,

making the reproduction of these methods difficult. The second is a lack of understanding of

the potential benefits of using hybrid data over conventional process or input-output data –

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although some literature can be found on the subject (Crawford, 2008; Lenzen, 2000).

Finally, there are no automated tools or software that would allow these methods to be

easily used by non-hybrid LCA specialists.

Throughout this paper, it is argued that conventional process and input-output

analysis represent the two ends of a spectrum which includes all hybrid methods (see

Figure 2 for illustrative purpose). Defining hybrid methods using the terms ‘process-based’

and ‘input-output-based’ is therefore misleading as it might not fully reflect the

hybridisation method used, i.e. it does not provide sufficient information to locate it within

the spectrum. Additionally, several hybrid methods are referred to as ‘input-output-based’

by their original authors (Joshi, 1999; Treloar, 1997) – a term reproduced by others using

distinctively different methods, thus adding to the confusion.

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Figure 2 The hybrid LCI method spectrum

A first step in addressing this issue is to clarify the naming conventions for the four

main hybrid LCI methods found in the literature. The following definitions are proposed (see

Figure 3 for a graphic representation of each method, and Appendix B for the details of each

method’s mathematical form):

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1. Tiered (Figure 3a): Combines input-output and process coefficients in order to

expand the boundaries of the system analysed and include information that would

typically not be represented in a conventional process analysis or EEIOA.

This method was first developed by Bullard et al. (1978), based on earlier work such as

that by Bullard and Herendeen (1975). Here, as defined by Heijungs and Suh (2002), the

boundaries between process and input-output data are defined by the user and thus they

may vary significantly – with some assessments using coefficients derived from input-output

tables to represent all inputs associated with the production processes while use and end-

of-life stages are modelled using process data (as used by Changbo et al. (2012) to assess a

biomass gasification system, for example). Alternatively, an extensive process analysis may

be undertaken first, and input-output coefficients are strictly used to account for inputs

and/or outputs associated with upstream processes for which no process data is available.

This approach was used by Zhai and Williams (2010) in the assessment of photovoltaic

systems, for example.

2. Path Exchange (PXC) (Figure 3b): Involves the mathematical disaggregation of an

input-output matrix, thus enabling the identification and modification of mutually

exclusive pathways – the sum of which represents the entire matrix.

First proposed by Treloar (1997), and more recently formalised by Lenzen and

Crawford (2009), the aim here is to analyse specific goods and services by tailoring their

supply chains with the use of available process data, while maintaining system boundary

completeness. This approach was used by Baboulet and Lenzen (2010) to assess the

performance of a university, enabling the tailoring of certain aspects of the supply chain to

represent different scenarios.

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3. Matrix Augmentation (Figure 3c and Figure 3d): Involves the direct modification of

the input-output matrix to create one or several additional ‘sectors’ of the economy. It

can either focus on dividing an existing sector into sub-sectors, or on creating a new,

theoretical sector.

First developed by Joshi (1999), the objective of this method is to assess a specific

good or service, either existing within a larger sector (for example, the study of wind power

by Wiedmann et al. (2011)), or which may not yet be included in input-output tables (as for

the study of biofuels in Australia by Malik et al. (2015)).

4. Integrated (Figure 3e): Integrates process and input-output data within a single

matrix framework, using a set of vectors referred to as upstream and downstream cut-off

matrix to link the two matrices. These vectors are used to represent inputs from the input-

output matrix into a process (upstream cut-off matrix), and sales of goods and services to

input-output sectors (downstream cut-off matrix).

Originally developed by Suh and Huppes (2000), this method has the advantage of

compiling process and input-output data into a common framework. This approach has

been used by Acquaye et al. (2012b) to assess the potential impact of biofuels on the British

economy, based on a range of scenarios.

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Figure 3 Graphic representation of different hybrid methods; (a) Tiered, (b) Path Exchange, (c) Matrix Augmentation - sector disaggregation, (d) Matrix Augmentation – new sector, (e) Integrated

5. Description of hybrid life cycle inventory methods

Every hybrid LCI method currently in use was theorised at least 15 years ago –

almost 40 years ago in the case of Tiered hybrid (Bullard et al., 1978). This section describes

each of these methods in further detail, highlighting their relative benefits and limitations,

and reviewing a selection of previous studies to highlight variations in their application. The

methods and terminologies relative to process analysis and EEIOA have been discussed in

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detail within the literature over many decades (Hendrickson et al., 1998; Kitzes, 2013; Suh

and Huppes, 2005) and are therefore not included here, but are essential components of

any hybrid LCI method.

5.1. Tiered hybrid

A Tiered hybrid analysis is based on a process analysis framework, but uses both

process and input-output data. Originally, it was proposed for process data to only be used

for ‘atypical products’, which could not easily fit in to any sector of the economy, while

input-output data could be used for all inputs typical of a sector (Bullard et al., 1978). It was

then proposed that process data could be used for prominent downstream processes where

specificity might be useful to the assessment, as well as the use and end-of-life phases

where input-output data would not be available (Suh and Huppes, 2005).

In essence, a Tiered hybrid analysis combines input-output and process data within a

process analysis framework to reduce the truncation error of a pure process analysis. In

practice, both types of data are often added in an ad-hoc manner. Scenarios of application

range from studies exclusively using input-output data except for the use and end-of-life

phases, to studies which use input-output data strictly for those inputs to the system for

which no process data is available.

There are two main drawbacks with the use of a Tiered hybrid analysis. First, it

follows the same method as a process analysis, therefore requiring the definition of a

system boundary and selection of processes to be represented by process data and those

represented by input-output data. Fundamentally, this means that some level of truncation

is still likely as the system boundary might exclude some processes, and the perceptions as

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to what the most prominent processes are, might be incorrect. Secondly, there is a risk of

double counting where the boundaries between process and input-output data are not

clearly defined (Suh and Huppes, 2005). To avoid double-counting, all aspects of the supply

chain covered by process data should be subtracted from the input-output matrix before

this data may be used. This process itself can be quite difficult without a fully disaggregated

input-output model.

Reviewing the literature, we found that studies using Tiered hybrid analysis are

applied in a variety of ways by researchers – sometimes simply using input-output derived

coefficients for material inputs for which no process data is available. An assessment of

what these coefficients may cover, or what other steps may be missing from the system

under study is not often performed. Additionally, we found a significant lack of consistency

in the names used to describe the method – sometimes not being referenced at all. To

illustrate these variations, two studies are reviewed below.

In a recent study, Ercan and Tatari (2015) set out to compare conventional public

transportation buses using fossil fuels with buses using alternative fuels via a hybrid

analysis. In doing so, the authors quote using an ‘input-output (IO)-based hybrid life cycle

assessment (LCA) model’. The study focuses on six types of fuel sources: diesel, hybrid,

compressed natural gas, liquid natural gas, 20% biodiesel and electric. A system boundary is

defined by the authors, which includes the manufacture of the buses themselves, the

necessary infrastructure to support their use, the variation in maintenance and of course

the fuel production and use. The authors then proceed to collect specific information

regarding the requirements of each aspect of the life cycle under study. Input-output

coefficients extracted using Carnegie Mellon University’s Economic Input-Output Life Cycle

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Assessment (EIO-LCA) method (CMU, 2003) are used to model the impacts associated with

the supply chain of each element considered, while the combustion of fuels itself is

modelled using specific US EPA emission factors. Although the input-output coefficient

would cover the entire upstream supply chain, the selection of foreground processes (e.g.

bus manufacture and fuel production) is likely to include some level of truncation. For

instance, non-physical flows such as supporting services are not included in the model.

Additionally, the reference to an input-output-based hybrid approach here is misleading, as

the method applied appears to fall under a process analysis framework.

A second example is a study from Zhang et al. (2015), which compares two types of

hydropower technologies. Here the authors report using a hybrid analysis without specifying

the type of method used – which can only be inferred from the details in the paper. A

planned hydropower station for which two technologies of dams were designed is used as a

case study. A system boundary is drawn, and process or input-output data are used based

on process data availability. For instance, the operation and maintenance of the dam is

modelled with input-output data, while transportation and construction is analysed using

process data. Some steps are explicitly excluded from the analysis, while some exclusions

are not discussed (e.g. non-physical flows) – again raising the question of truncation.

As can be seen from a brief review of typical examples, the method used for, and

understanding of the application of a Tiered hybrid analysis can vary significantly from one

study to another – thus affecting the level of confidence that can be placed in the results of

these assessments and the potential comparability between studies. While some authors

provide detailed background information on the method used and the steps followed to

assess and address the limitations of their analysis, many simply use third-party derived

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input-output coefficients to represent a selection of inputs for which no process data is

available.

5.2. Path Exchange (PXC) hybrid

The PXC method was first proposed by Treloar (1997), and later formalised by

Lenzen and Crawford (2009), building on the work of researchers from the previous decade.

In the mid-1980s, Miller and Blair (2009) reported a mathematical equation that could be

used to disaggregate the Leontief inverse formula into a series of stages starting from zero

and theoretically going to infinity. Here, every term of the resulting equation represents a

level of the supply chain. Building on this, Treloar (1997) introduced the idea of further

disaggregating this equation into discrete nodes, which, when combined, would represent

every input from every sector at every level of the supply chain.

The first step of the PXC method is to mathematically disaggregate the input-output

matrix into a series of mutually exclusive nodes, each representing a good or service

provided by a particular input-output sector. Node to node connections represent a

transaction between input-output sectors, i.e. the purchase of a good or service from one

sector by another. A series of nodes, corresponding to a chain of transactions leading to the

sector being assessed, is referred to as a pathway. Specific nodes are then modified using

process data that corresponds with the particular transaction. The modifications can affect

either the value of the transaction, if identified as different for the particular good or service

under study, or the environmental flow associated with the transaction, if specific process

data is available.

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From such an assessment, data within specific discrete pathways may be replaced

with process data specific to the product under study without affecting the overall matrix

(Baboulet, 2009; Baboulet and Lenzen, 2010; Crawford, 2008, 2011; Lenzen and Crawford,

2009). The result is the entire supply chain of a sector modified to represent a specific

product or process.

Although this method can theoretically be applied manually, the sheer number of

data to analyse can make it time consuming in practice. The complexity of this method and

amount of data to be manipulated is one reason explaining why its use hasn’t yet become

common practice.

The PXC method is regularly referred to in publications using hybrid methods, but its

application is rare and often limited to the group of researchers behind its development.

Because of this, its application is homogenous within the literature. Two approaches have

been found, the first applying the method in its most rudimentary form to a system and the

other relying first on hybrid coefficients developed separately using the PXC method. Two

publications were selected to illustrate the two approaches (Baboulet and Lenzen, 2010;

Rauf and Crawford, 2015).

Baboulet and Lenzen (2010) published the first application of the PXC method

following its formalisation by Lenzen and Crawford (2009). Throughout this study they refer

to the type of hybrid assessment as the Path Exchange method. Here, the authors set out to

assess the energy use and greenhouse gas emissions associated with the activities

undertaken by a university – directly and indirectly. The assessment also aims to review the

potential benefits or impacts of a modification in University spending. As prescribed by the

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method (Lenzen and Crawford, 2009), the first step in the study was to calculate the total

environmental flows associated with the university’s activities, using a conventional EEIOA.

A structural path analysis (SPA) was then undertaken to identify the top ranking nodes for

the indicators assessed and the environmental flows associated with each. Using the PXC

method, specific process data was used to modify pathways to reflect the implementation

of various scenarios in university spending. This allowed the authors to not only assess the

university’s activities in a more detailed manner, but also to provide useful feedback in how

to reduce greenhouse gas emissions related to changes in the university’s supply chain.

Another approach to the PXC method is to use hybrid coefficients, such as those

developed by Treloar and Crawford (2010), which has been found in a number of

publications (Crawford and Stephan, 2013; Rauf and Crawford, 2015; Stephan et al., 2012,

2013a). Throughout these applications, authors tend to refer to the hybrid method as

‘input-output-based hybrid’. In the most recent example, Rauf and Crawford (2015)

developed a model for assessing how the service life of a building influences the energy

consumed across its life cycle. In this assessment, the authors use a series of hybrid energy

coefficients developed with the use of the PXC method for each specific material used

during the construction and subsequent use of the building (Crawford, 2011; Treloar and

Crawford, 2010). These coefficients are multiplied by the physical material quantities within

the building. In order to ensure a complete system boundary, the authors calculate what

they call the ‘input-output remainder’ which represents all inputs that are not covered by

the physical material quantities. The calculation of the remainder involves several steps.

First, using a disaggregated input-output model, a SPA is conducted for the relevant

construction sector and the pathways covered by the hybrid coefficients are identified. The

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total energy requirement of these pathways is then subtracted from the total energy

requirement of the construction sector in order to obtain the remainder. This value is

multiplied by the total cost of constructing the building and added to the value obtained

using the hybrid coefficients to determine the total hybrid energy requirement associated

with the building. This approach simplifies the hybridisation method by creating a pre-

compiled database of hybrid coefficients, thus allowing an assessment to be performed in a

much more efficient manner. These coefficients can then be periodically updated as new

process or input-output data becomes available.

In these two previous studies, we can see that the way in which the authors have

applied the PXC method appears to vary quite significantly. However, the process of

hybridisation in itself is actually very similar, which can be explained by the fact that the co-

authors of both papers were involved in the development of the PXC method. The

development of a more comprehensive database of hybrid coefficients, that is made

available to practitioners, could be a pathway to further disseminate the PXC method.

Additionally, it would be valuable to create a more automated approach for the integration

of available process data within the input-output model to reduce the time and complexity

associated with this process.

5.3. Matrix Augmentation

This method, also often referred to as input-output-based hybrid, was originally

proposed by Joshi (1999). It has, since then, been widely used in the literature (Malik et al.,

2015; Qiuhong et al., 2014; Wiedmann et al., 2011; Zafrilla et al., 2014).

25

As described in the methodological literature (Baboulet, 2009; Joshi, 1999; Suh and

Huppes, 2005), there are two possible applications of the method, both of which result in

the same outcome – the addition of a ‘new’ sector to the input-output matrix. First, it can

be used to disaggregate a sector into sub-sectors for a more specific assessment, based on

data such as economic output. This is referred to by Joshi (1999) as Model III. In the first

instance, all inputs are calculated proportionally to the original sector. They are then

modified using the available process data, while inputs for which process data is unavailable

are kept as is. Outputs are often kept in the same sale structure as the original sector.

The second application is to add one theoretical sector to the matrix. This is referred

to by Joshi (1999) as Model II. This method is applicable in particular to new or emerging

technologies or sectors. Here, process data is used to simulate the physical input

requirements of the new sector, while non-physical inputs may be covered using input-

output data from a similar sector of the economy. As with Model III, outputs are often given

the same sale structure as the original sector. For instance, in a study on biofuels, Malik et

al. (2015) use the same sale structure as fossil fuels, assuming that they would be used in a

similar way within the economy.

Both of these applications set out to resolve the aggregation error found in

conventional EEIOA. The main drawback is that the Matrix Augmentation approach works

within the main input-output table, thus every modification of the matrix is potentially

reverberated across every tier of the supply chain. Additionally, the aggregation error is only

partially dealt with, as users often assume a similar output structure as an existing sector. It

is also only addressed at the first level of the assessment, and aggregation will still exist in

relation to other sectors throughout the matrix.

26

Two peer-reviewed publications using these two applications were reviewed as

examples, one using Model III (Li et al., 2012; Qiuhong et al., 2014), and one using Model II

(Hou et al., 2014; Malik et al., 2015), to show some of the variations found in their

application.

Li et al. (2012) aimed to assess the production of electricity from wind technologies

in China, using a method they referred to as ‘input-output-based hybrid’. In this assessment,

the national economic output of wind power is used to disaggregate the electricity sector of

the Chinese input-output table into a sector representing wind power and a sector

representing all other sources of electricity. A first step in the assessment is to define the

ratio of representativeness of wind power within the electricity sector. This is used to split

the sector under study, keeping the proportion of inputs/outputs the same for the two

newly created sectors. A process-based life cycle inventory (LCI) of a wind turbine

representative of the average turbine installed in China is then used to adapt the newly

created ‘wind power’ sector. All LCI inputs were collected as physical values and converted

to monetary values based on the China Price Yearbook (NDRC, 2008). These are inserted in

lieu of the original inputs calculated. Additionally, sectors which are reported by the authors

as irrelevant are set to zero – e.g. the tobacco sector, and their inputs are reverted to the

‘all other sources of electricity’ sector. The main output of this analysis is to virtually create

an input-output sector that is representative of the wind power industry, including inputs

adapted from process and input-output data.

Hou et al. (2014) applied the Matrix Augmentation method to assess the potential

impacts of the sediment remediation process that took place in the land redevelopment

associated with the construction of the London Olympic Park, prior to the 2012 Olympic

27

Games. The study focuses on the management of contaminated sediments in waterways

that were proposed to be revived as part of the redevelopment project, and uses an

approach referred to as ‘input-output-based hybrid’. The planning options assessed were

either to take no action or to dredge the sediment, so that transport could be resumed on

the waterways. A large proportion of sediment needed to be dredged and was identified as

contaminated, making it a hazardous waste. Two management options were modelled;

either sending the sediment to landfill ‘as is’, or treating it using a soil washing process.

Here, two new industry sectors were added to the matrix to represent the two potential

remediation projects. Both were based on the existing remediation industry sector. Their

inputs were then adapted to represent each process, using ecoinvent data, project specific

data, and other publicly available sources of information. Applying this method allowed the

authors to compare two processes that are part of the same overarching sector by

estimating the inputs of these processes and linking them to the whole economy.

In these two cases, approaches with varying levels of robustness were used by

different authors, both quoting Joshi (1999) as the basis of their method. Two ways in which

input-output matrices have been augmented to ‘create’ sectors specific to the assessment

are evident. Often, the newly created sector is entirely proportional to the original sector

(Qiuhong et al., 2014), thus limiting the potential benefits of the Matrix Augmentation

method. The method therefore has some clear shortcomings, as the sector disaggregation

has no effect on the absolute results, making the method similar to a Tiered hybrid.

5.4. Integrated hybrid

The Integrated hybrid method was first proposed in the early 2000s (Heijungs and

Suh, 2002; Suh and Huppes, 2000, 2005; Suh et al., 2004). In this method, process data is

28

represented as a technology matrix and then connected to the input-output table. Thus, it

attempts to solve the analysis strictly via matrix computations. The connections between

the process and input-output matrices are vectors of inputs and outputs, which are referred

to as the upstream and downstream cut-offs, respectively (Acquaye et al., 2011b; Baboulet,

2009; Suh and Huppes, 2005; Suh et al., 2004). In effect, their role is to counteract the

systematic truncation of a process analysis. The upstream cut-off is used to represent inputs

to the system under study that arise from economic sectors for which no process data is

available. The downstream cut-off represents the sales of goods and services from the

technology matrix to other economic sectors.

The method has been used in a number of studies since it was first proposed

(Acquaye et al., 2011b; Bush et al., 2014; Jiang et al., 2014; Liu et al., 2012; Rodriguez-Alloza

et al., 2015; Wiedmann et al., 2011). A common way to describe and explain its

mathematical representation is that the original input-output table is augmented with a

process matrix. The system is then rebalanced by subtracting the upstream and downstream

cut-offs. One of the main concerns with this method is the potential for double counting.

This is due to the introduction of process data that no longer sums with the rest of the

input-output table to a complete description of the entire economy. Parts of the economic

system may be described twice in the resultant table, introducing the potential for double

counting which is harder to avoid than in the case of Tiered or PXC hybrid. In addition to

this, the compilation of the cut-off matrices has been reported as being highly data and time

intensive (Baboulet, 2009).

29

Our analysis suggests that it is a method which is applied in a fairly homogeneous

manner, as exemplified by the two studies reviewed below (Bush et al., 2014; Inaba et al.,

2010).

Inaba et al. (2010) used an Integrated hybrid analysis to assess alternative waste

management practices for household waste in Japan. Two scenarios were analysed, the first

one under which all combustible waste is incinerated, the second where food waste is

biogasified and the rest incinerated. In this assessment, process data is collected to

represent both alternative management systems, and entered into a technology matrix. The

process data is then linked to the relevant input-output table by the creation of a vertical

vector of inputs, as graphically represented in Figure 3 – referred to as the upstream cut-off.

Values entered within the upstream cut-off vector are based on inputs of the corresponding

sector within the input-output table. Inputs identified as covered by process data are

subtracted from the original sector and the result is added to the upstream cut-off vector

(see Figure 3). This assessment allows the authors to compare two waste management

systems and analyse the results in detail, while keeping a complete system boundary and

countering the limitations of a conventional EEIOA.

More recently, Bush et al. (2014) used an Integrated hybrid analysis to assess

electricity micro-generation systems and their carbon payback period under different

installation conditions. The assessment focuses on photovoltaic and micro-wind. Here, a

process matrix is developed which includes all process data collected. This matrix is linked to

the input-output matrix via two sets of vectors, the upstream and downstream cut-off, in a

similar way to Inaba et al. (2010). The downstream cut-off is a horizontal vector of output

30

which represents the sales of goods and services represented within the process matrix to

other sectors of the economy.

The main variation in the application of the Integrated hybrid method is in its use of

the upstream and downstream cut-off vectors (see Figure 3). Many authors argue that the

downstream cut-off vector has a minimal effect on the result, while potentially requiring

significant effort to be determined and is thus often not included. In response to this

simplification, it has been argued that although it is true that this vector will often have

negligible effect on the results, there are cases where it would (Suh, 2006) and thus should

at least be considered, and if excluded, well justified by the authors.

6. Discussion and future outlook

In the course of this study, we analysed the currently available methods of

hybridising process and input-output data for the purpose of creating a LCI. From this

analysis, it is apparent that all methods are found within a spectrum that is between process

and input-output analysis – therefore we argue that it is misleading (and limiting) to label a

method as ‘process-based’ or ‘input-output-based’, and more relevant and useful to refer to

a specific method of process and input-output data integration. Many of the studies found

did not report the type of method used, or used similar terms to refer to different methods.

In particular, studies using the Matrix Augmentation or PXC method consistently refer to

their method as ‘input-output-based’ when both methods vary significantly in their

framework.

We found significant inconsistencies in the way certain methods were applied, as

described in Section 5. In particular, the Tiered hybrid method often appeared to be used in

31

an ad-hoc manner. Over 40% of studies using this method did not quote what method was

used, and only 24% clearly reported using a Tiered hybrid approach. Over the course of our

analysis, ten different method names were given to studies that appear to use a Tiered

hybrid method (see Table 2). In contrast, the Integrated hybrid method is used in a fairly

consistent way across the studies we found. This may be due to its clearly defined

mathematical framework, reported in methodological publications (Suh, 2002; Suh and

Huppes, 2005, 2009), as well as in case studies (Acquaye et al., 2012b; Dadhich et al., 2015;

Wiedmann et al., 2011). Similarly, studies using the PXC method appear to apply it in a

consistent manner. Here, again, the framework of the method is clearly defined (Lenzen and

Crawford, 2009) but it is also worth noting that most studies are performed by a handful of

authors which inherently leads to a more consistent use of the method.

This study highlights a clear need for LCA researchers and practitioners to be

extremely transparent in the way they report the method used in their analyses, both in

how they refer to their method and how they describe it. At a minimum, every scientific

publication utilising a hybrid LCI method should refer to the exact method of hybridisation

used. This would help reduce confusion in the LCA literature and enable greater

comparability between studies.

As one of the main aims of this paper, we proposed a consistent framework for

classifying and defining the four broad types of hybrid analysis identified in the literature.

Referring to and using an accepted and standardised set of definitions for the type of LCI

method used in LCA studies will help provide consistency within the growing body of work

on hybrid LCI, making it more accessible to researchers and practitioners.

32

Additionally, the four methods themselves should be normalised to allow for clearer

and more comparable assessments. Our analysis suggests that the PXC and Integrated

hybrid methods are the most fit for normalisation, as a strict framework is already in place

to apply these methods. Additionally, based on our analysis of all four hybrid methods,

these two provide the most comprehensive approach for hybridising process and input-

output data.

Life cycle assessments based on a hybrid LCI have the potential to bring value to the

life cycle assessment field by providing a more comprehensive analysis of a product system.

It is therefore clear that hybrid LCI methods should be promoted to the wider LCA

community. This paper also aimed to provide recommendations to facilitate the uptake of

these methods by researchers and practitioners. Several steps appear crucial to increase the

uptake of these methods. First and foremost, their added-value, when compared to

conventional process analysis, must be discussed further in scientific publications. Although

a number of studies have highlighted the potential shortcomings of process analysis

(Crawford, 2008; Lenzen, 2000), more practical examples need to be published to show the

variation in results between process and hybrid analysis. Disseminating this knowledge

further will certainly help the uptake of these methods.

Additionally, data must be made available in a format that allows researchers and

practitioners to conduct a hybrid analysis. Allowing users of the most common LCA

software, such as Simapro and GaBi, to use a hybrid LCI method such as the Integrated or

PXC method in their assessment is also a critical step towards the uptake of hybrid LCI

methods. Addressing the time and complexity involved in the hybridisation process should

33

also be a priority. To this end, Crawford et al. (2017a); Crawford et al. (2017b) have

proposed an approach for automating this process for the PXC method.

7. Conclusion

There has been increasing methodological work on developing suitable hybrid LCI

methods in the past two decades, but their use is still limited in comparison to conventional

process LCI. There are a number of reasons for this slow uptake. The first is a lack of

standardisation in the methods used, as methods are mainly discussed in scientific

publications, but a formal step by step definition of each method is not yet part of LCA

Standards documentation. The second reason is a lack of tools and software allowing the

general use of hybrid methods by LCA practitioners. The automation of the process and

input-output data integration task would make the methods more accessible to

practitioners which would in turn help its uptake. One valuable step in this direction would

be the development of a database of hybrid coefficients, building on the existing work of

Treloar and Crawford (2010). High quality and detailed process and input-output databases

are now available, and their hybridisation would create a highly valuable source of

information for researchers and practitioners alike. Complex methods such as the PXC

method would greatly benefit from a model capable of more efficiently integrating process

and input-output data at the node and pathway level. An analysis of the sample of literature

we considered clearly suggests that there is a community of researchers capable of

implementing any of the existing hybridisation methods, however formalising these

methods within a software model for use by a broader, often less educated, group of users

would extend the boundaries of this practice, and perhaps show its potential benefits to the

wider LCA community.

34

Acknowledgement

This research was supported by the Australian Research Council's Discovery Projects funding

scheme (project number DP150100962).

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42

Appendix A – Summary of publications analysed

Author/s Year Country of 1st author affiliation

Focus of assessment

Name of hybrid method quoted by authors

Our interpretation

T MA I PXC

Baboulet and Lenzen (2010)

2010 AU University activities

Path Exchange X

Bilec et al. (2010) 2010 US Buildings - X

Bright et al. (2010)

2010 NO Biofuel - X

Goggins et al. (2010)

2010 IE Concrete Process-based X

Harto et al. (2010)

2010 US Fuels Additive X

Hassan (2010) 2010 US Titanium coating - X

Inaba et al. (2010)

2010 JP Waste management

Product equilibrium X

Mattila et al. (2010)

2010 FI Industrial activity Tiered X

Mo et al. (2010) 2010 US Water - X

Morris and Matthews (2010)

2010 US Products - X

Nakamura and Yamasue (2010)

2010 JP Waste management

- X

Peters et al. (2010a)

2010 AU Food Tiered X

Peters et al. (2010b)

2010 AU Food Tiered X

Zhai and Williams (2010)

2010 US PV systems Additive X

Acquaye et al. (2011a)

2011 IE Buildings Stochastic X

Acquaye et al. (2011b)

2011 UK Biodiesel Integrated X

Arvesen and Hertwich (2011)

2011 NO Wind power - X

43

Berners-Lee et al. (2011)

2011 UK Business operations

- X

Cass and Mukherjee (2011)

2011 US Highway construction

- X

Deng et al. (2011) 2011 US Laptop Economic balance X

Ewing et al. (2011)

2011 US Marine freight - X

Heinonen and Junnila (2011)

2011 FI Cities Tiered X

Koskela et al. (2011)

2011 FI Economy imports - X X

Mo et al. (2011) 2011 US Water supply and treatment

Input-output-based X X

Singh et al. (2011)

2011 NO Energy - X

Virtanen et al. (2011)

2011 FI Food - X

Weinzettel and Kovanda (2011)

2011 CZ Economy - X

Wiedmann et al. (2011)

2011 UK Wind power Integrated; Input-output-based

X X

Acquaye et al. (2012b)

2012 UK Biofuels Integrated X

Acquaye et al. (2012a)

2012 UK Photovoltaic system

Integrated X

Aye et al. (2012) 2012 AU Buildings Input-output-based X

Chang et al. (2012)

2012 US Buildings Process-based X

Changbo et al. (2012)

2012 CN Biomass gasification

Additive X

Glew et al. (2012) 2012 UK Mattresses Integrated X

Li et al. (2012) 2012 UK Wind power Input-output-based X

Liu et al. (2012) 2012 TW Taiwan’s economy Input-output-based X

44

Mo (2012) 2012 US Water supply and treatment

Input-output-based X X

Nakamura et al. (2012)

2012 JP Recycling - X

Stephan et al. (2012)

2012 BE Buildings Input-output-based X

Arvesen (2013) 2013 NO Energy Tiered X

Dong et al. (2013) 2013 CN Industrial precinct Tiered X

Famuyibo et al. (2013)

2013 IE Buildings Process-based X

Kovanda and Weinzettel (2013)

2013 CZ Economy - X

Lee and Ma (2013)

2013 TW Products Integrated X

Lin et al. (2013) 2013 CN City - X

Lindner et al. (2013)

2013 UK Electricity - X

Stephan et al. (2013a)

2013 BE Buildings Input-output-based X

Stephan et al. (2013b)

2013 BE Buildings Input-output-based X

Arvesen et al. (2014)

2014 NO Wind power Tiered X

Aurangzeb et al. (2014)

2014 US Products - X

Bush et al. (2014) 2014 UK Energy Integrated X

Feng et al. (2014) 2014 CN Energy Integrated X

Font Vivanco et al. (2014)

2014 NL Electric cars - X

Hou et al. (2014) 2014 UK Process Input-output-based X

Igos et al. (2014) 2014 LU Energy - X

Jiang et al. (2014) 2014 CN Products Integrated X

Malik et al. (2014)

2014 AU Energy - X

45

Meylan et al. (2014)

2014 CH Waste - X

Moran et al. (2014)

2014 NO Minerals - X

Noori et al. (2014)

2014 US Wind energy Economic input-output-based

X

Omar et al. (2014)

2014 MY Products - X

Onat et al. (2014a)

2014 US Buildings Tiered X

Onat et al. (2014b)

2014 US Buildings - X

Qiuhong et al. (2014)

2014 CN Products Input-output-based X

Stephan and Stephan (2014)

2014 BE Buildings Input-output-based X

Vasan et al. (2014)

2014 US Electronic products Economic balance X

Wan Omar et al. (2014)

2014 AU Construction materials

- X

Wang and Yuan (2014)

2014 US Process - X

Yang et al. (2014) 2014 SE Solar water pumping

- X

Yao et al. (2014) 2014 US PV system - X X

Zafrilla et al. (2014)

2014 ES Energy Tiered X

Alvarez and Rubio (2015)

2015 ES Wood pallet MC3 X

Bawden and Williams (2015)

2015 US Buildings Additive EIO LCA X

Bouman et al. (2015)

2015 NO Energy Tiered X

Caro et al. (2015) 2015 IT Nation Hybrid input-output X

Dadhich et al. (2015)

2015 UK Construction Integrated X

46

Dixit et al. (2015) 2015 US Products Input-output-based X

Ercan and Tatari (2015)

2015 US Transport Input-output-based X

Hertwich et al. (2015)

2015 NO Energy Integrated X

Hong et al. (2015) 2015 CN Buildings Input-output-based X

Igos et al. (2015) 2015 LU Policy Hybrid input-output X

Jang et al. (2015) 2015 KR Buildings - X

Kjaer et al. (2015) 2015 DK Corporation; Transport

Tiered; Integrated; X X X

Lake et al. (2015) 2015 UK Supply chains - X

Malik et al. (2015)

2015 AU Energy Input-output-based X

Ottelin et al. (2015)

2015 FI Buildings Tiered X

Pairotti et al. (2015)

2015 IT Diet - X

Palma-Rojas et al. (2015)

2015 BR Biofuels Integrated X

Rauf and Crawford (2015)

2015 AU Buildings Path Exchange X

Rodriguez-Alloza et al. (2015)

2015 ES Products Input-output-based X

Ryen et al. (2015) 2015 US Consumer electronics

Consumption-weighted LCA

X

Shahabi et al. (2015)

2015 AU Desalination process

Hybrid input-output X

Vendries Algarin et al. (2015)

2015 US Electricity Input-output-based X

Wang et al. (2015)

2015 CN Energy Tiered X

Watanabe et al. (2015)

2015 BR Biofuels Input-output-based X

Zhang et al. (2015)

2015 CN Energy - X

47

Zhao and Tatari (2015)

2015 US Vehicle use Economic input-output-based

X

Note: T: Tiered; MA: Matrix Augmentation; I: Integrated; PXC: Path Exchange

48

Appendix B – Mathematical forms

This section supports the hybrid classification proposed throughout the main body of

the paper by providing the mathematical equations relevant to each hybrid method, as well

as pure input-output and process analysis, using a common notation. All equations are

adapted from (Heijungs and Suh, 2002), (Lenzen and Crawford, 2009), (Joshi, 1999) and

(Suh, 2004).

B.1 Input-output analysis

The mathematical form of a pure input-output analysis has been described in

multiple publications over the years. For assessing m environmental burdens and

requirements associated with a specific sector 𝑆𝑆 of an economy described by n sectors,

equation (B.1) may be applied.

𝑞𝑞𝑠𝑠 = 𝑅𝑅(𝐼𝐼 − 𝐴𝐴)−1𝑦𝑦𝑠𝑠 (B.1)

Where:

𝑞𝑞 = the vector of total environmental burdens and requirements of the sector 𝑆𝑆,

of dimension 𝑚𝑚 × 1;

𝑅𝑅 = the matrix of 𝑚𝑚 environmental burdens and requirements associated with all

𝑛𝑛 sectors of the economy, of dimension 𝑚𝑚 × 𝑛𝑛;

𝐼𝐼 = the identity matrix of dimension 𝑛𝑛 × 𝑛𝑛;

𝐴𝐴 = the input-output matrix of dimension 𝑛𝑛 × 𝑛𝑛; and

𝑦𝑦 = the vector of final demand for the sector 𝑆𝑆, of dimension 𝑛𝑛 × 1.

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B.2 Process analysis

In comparison with input-output analysis, the mathematical form of a pure process

analysis has seldom been described in the literature. For assessing t environmental burdens

and requirements associated with a specific unit process 𝑈𝑈 of a database of z processes,

equation (B.2) may be applied.

𝑞𝑞𝑢𝑢 = 𝑅𝑅�𝑇𝑇−1𝑦𝑦�𝑢𝑢 (B.2)

Where:

𝑞𝑞 = the vector of total environmental burdens and requirements of the process

𝑈𝑈, of dimension 𝑡𝑡 × 1;

𝑅𝑅� = the matrix of 𝑡𝑡 environmental burdens and requirements associated with all

𝑧𝑧 processes of the database, of dimension 𝑡𝑡 × 𝑧𝑧;

𝑇𝑇 = the technological matrix of dimension 𝑧𝑧 × 𝑧𝑧; and

𝑦𝑦� = the vector of final demand for the process 𝑈𝑈, of dimension 𝑧𝑧 × 1.

B.3 Hybrid analysis

The mathematical forms of the various hybrid LCI methods have been described in

various publications. We attempt to summarise their mathematical form using a consistent

notation, to further clarify the way in which they vary. We differentiate between input-

output, process and hybrid data within the equations. We refer to sectors 𝑆𝑆 for input-output

data, unit process 𝑈𝑈 for process data, and hybrid component 𝐻𝐻 for hybridised data. It

should be noted that by its very nature, a hybrid LCI will consider a fixed set of

environmental burdens and/or requirements and so the range of t environmental burdens

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and requirements covered by the process matrix (𝑅𝑅�) will be equivalent to the range of m

environmental burdens and requirements covered by the input-output matrix (𝑅𝑅). Thus,

from this point forward, the range of environmental burdens and requirements considered

is referred to as m.

B.3.1 Tiered hybrid

A Tiered hybrid (see Figure 3 (a)) can be simply described as the sum of input-output

and process components, as per equation (B.3).

𝑞𝑞h = 𝑞𝑞𝑠𝑠 + 𝑞𝑞𝑢𝑢 (B.3)

Using equations (B.1) and (B.2), equation (B.3) can be expanded into equation (B.4)

below.

𝑞𝑞h = 𝑅𝑅(𝐼𝐼 − 𝐴𝐴)−1𝑦𝑦s + 𝑅𝑅�𝑇𝑇−1𝑦𝑦�u (B.4)

The two components of equation (B.4) can then be combined as shown in equation

(B.5) below.

𝑞𝑞h = (𝑅𝑅� 𝑅𝑅) �𝑇𝑇 00 𝐼𝐼 − 𝐴𝐴�

−1�𝑦𝑦�u𝑦𝑦s� (B.5)

Where:

51

𝑞𝑞 = the vector of total environmental burdens and requirements of the hybrid

component 𝐻𝐻 being analysed, of dimension 𝑚𝑚 × 1; 𝑞𝑞𝑠𝑠 and 𝑞𝑞𝑢𝑢 its input-output

and process components;

𝑇𝑇 = the technological matrix of dimension 𝑧𝑧 × 𝑧𝑧;

𝐼𝐼 = the identity matrix of dimension 𝑛𝑛 × 𝑛𝑛;

𝐴𝐴 = the input-output matrix of dimension 𝑛𝑛 × 𝑛𝑛;

𝑦𝑦𝑠𝑠 and 𝑦𝑦�𝑢𝑢 = the vector of final demand for the sector 𝑆𝑆 associated with the hybrid

component 𝐻𝐻, of dimension 𝑛𝑛 × 1, and the vector of final demand for the unit

process 𝑈𝑈 associated with the hybrid component 𝐻𝐻, of dimension 𝑧𝑧 × 1,

respectively; and

𝑅𝑅 and 𝑅𝑅� = the matrices of 𝑚𝑚 environmental burdens and requirements for all 𝑛𝑛

sectors of the economy and of dimension 𝑚𝑚 × 𝑛𝑛, and all 𝑧𝑧 processes of the

database of dimension 𝑚𝑚 × 𝑧𝑧, respectively.

B.3.2 Path Exchange

The Path Exchange method (see Figure 3 (b)) differs from other approaches in that it

starts with the disaggregation of the input-output and technological matrix, as can be seen

in equation (B.6) and (B.7), for the sector 𝑆𝑆 and unit process 𝑈𝑈, both corresponding to the

hybrid component 𝐻𝐻 being analysed. For clarity, all terms of equation (B.6) to (B.12) are

defined after equation (B.12).

𝑞𝑞s = 𝑅𝑅𝑦𝑦s + 𝑅𝑅𝐴𝐴𝑦𝑦s + 𝑅𝑅𝐴𝐴2𝑦𝑦s + ⋯+ 𝑅𝑅𝐴𝐴∞𝑦𝑦s (B.6)

52

𝑞𝑞u = 𝑅𝑅�𝑦𝑦�u + 𝑅𝑅�𝑇𝑇𝑦𝑦�u + 𝑅𝑅�𝑇𝑇2𝑦𝑦�u + ⋯+ 𝑅𝑅�𝑇𝑇∞𝑦𝑦�u (B.7)

Each stage unravelled in equation (B.6) and (B.7) can then be further disaggregated

into a series of pathways. This can be done in a similar manner for input-output (equation

(B.8) and (B.9)) and process data (equation (B.10) and (B.11)), provided that for the latter,

process data has been formatted in a matrix form. The disaggregation of the matrix shows

inputs of other sectors (𝑗𝑗,𝑘𝑘, 𝑙𝑙) to the sector 𝑆𝑆 under study, or of other unit processes

(𝑡𝑡, 𝑣𝑣,𝑤𝑤) into the unit process 𝑈𝑈 under study.

𝑞𝑞s = �𝑦𝑦𝑗𝑗𝑟𝑟𝑗𝑗�𝛿𝛿𝑗𝑗𝑠𝑠 + 𝐴𝐴𝑗𝑗𝑠𝑠 + (𝐴𝐴2)𝑗𝑗𝑠𝑠 + ⋯+ (𝐴𝐴∞)𝑗𝑗𝑠𝑠�𝑛𝑛

𝑗𝑗=1

(B.8)

𝑞𝑞s = �𝑦𝑦𝑗𝑗𝑟𝑟𝑗𝑗 �𝛿𝛿𝑗𝑗𝑠𝑠 + 𝐴𝐴𝑗𝑗𝑠𝑠 + �𝐴𝐴𝑗𝑗𝑗𝑗𝐴𝐴𝑗𝑗𝑠𝑠

𝑛𝑛

𝑗𝑗=1

+ ��𝐴𝐴𝑗𝑗𝑗𝑗𝐴𝐴𝑗𝑗𝑗𝑗𝐴𝐴𝑗𝑗𝑠𝑠

𝑛𝑛

𝑗𝑗=1

𝑛𝑛

𝑗𝑗=1

+ ⋯�𝑛𝑛

𝑗𝑗=1

(B.9)

𝑞𝑞u = �𝑦𝑦�𝑡𝑡�̃�𝑟𝑡𝑡(𝛿𝛿𝑡𝑡𝑢𝑢 + 𝑇𝑇𝑡𝑡𝑢𝑢 + (𝑇𝑇2)𝑡𝑡𝑢𝑢 + ⋯+ (𝑇𝑇∞)𝑡𝑡𝑢𝑢)𝑧𝑧

𝑡𝑡=1

(B.10)

𝑞𝑞u = �𝑦𝑦�𝑡𝑡�̃�𝑟𝑡𝑡 �𝛿𝛿𝑡𝑡𝑢𝑢 + 𝑇𝑇𝑡𝑡𝑢𝑢 + �𝑇𝑇𝑡𝑡𝑡𝑡𝑇𝑇𝑡𝑡𝑢𝑢

𝑛𝑛

𝑡𝑡=1

+ ��𝑇𝑇𝑡𝑡𝑡𝑡𝑇𝑇𝑡𝑡𝑡𝑡𝑇𝑇𝑡𝑡𝑢𝑢

𝑛𝑛

𝑡𝑡=1

𝑛𝑛

𝑡𝑡=1

+ ⋯�𝑧𝑧

𝑡𝑡=1

(B.11)

Every iteration of 𝐴𝐴𝑗𝑗𝑠𝑠 and 𝑇𝑇𝑡𝑡𝑢𝑢 for the sector 𝑆𝑆 and unit process 𝑈𝑈 corresponds to a

node in the supply chain (or process tree). The path exchange functions by assessing

correspondence between specific nodes in the input-output and process data. Let’s assume

53

that we are assessing a hybrid component 𝐻𝐻, which corresponds to the input-output sector

𝑆𝑆 and process 𝑈𝑈. After unravelling the supply chain and process tree, and analysing the

nodes, we find that 𝑦𝑦�𝑡𝑡 corresponds to 𝑦𝑦𝑗𝑗, �̃�𝑟𝑡𝑡 corresponds to 𝑟𝑟𝑗𝑗, 𝛿𝛿𝑡𝑡𝑢𝑢 corresponds to 𝛿𝛿𝑗𝑗𝑠𝑠, 𝑇𝑇𝑡𝑡𝑢𝑢

corresponds to 𝐴𝐴𝑗𝑗s, and 𝑇𝑇𝑡𝑡𝑡𝑡 corresponds to 𝐴𝐴𝑗𝑗𝑗𝑗.

The modified supply chain could then be summarised as in equation (B.12).

𝑞𝑞ℎ = 𝑦𝑦�𝑡𝑡�̃�𝑟𝑡𝑡 �𝛿𝛿𝑡𝑡𝑢𝑢 + 𝑇𝑇𝑡𝑡𝑢𝑢 + �𝑇𝑇𝑡𝑡𝑡𝑡𝐴𝐴𝑗𝑗𝑠𝑠

𝑛𝑛

𝑗𝑗=1

� + �𝑦𝑦𝑗𝑗𝑟𝑟𝑗𝑗 ���𝐴𝐴𝑗𝑗𝑗𝑗𝐴𝐴𝑗𝑗𝑗𝑗𝐴𝐴𝑗𝑗𝑠𝑠

𝑛𝑛

𝑗𝑗=1

𝑛𝑛

𝑗𝑗=1

+ ⋯�𝑛𝑛

𝑗𝑗=1

(B.12)

Where:

𝑞𝑞 = the vector of total environmental burdens and requirements of the hybrid

component 𝐻𝐻 being analysed, of dimension 𝑚𝑚 × 1;

𝑇𝑇 = the technological matrix of dimension 𝑧𝑧 × 𝑧𝑧;

𝐴𝐴 = the input-output matrix of dimension 𝑛𝑛 × 𝑛𝑛;

𝑦𝑦 and 𝑦𝑦� = the vector of final demand for the sector 𝑆𝑆 associated with the hybrid

component 𝐻𝐻, of dimension 𝑛𝑛 × 1, and the vector of final demand for the

process 𝑈𝑈 associated with the hybrid component 𝐻𝐻, of dimension 𝑧𝑧 × 1,

respectively;

𝑅𝑅 and 𝑅𝑅� = the matrices of 𝑚𝑚 environmental burdens and requirements for all 𝑛𝑛

sectors of the economy and of dimension 𝑚𝑚 × 𝑛𝑛, and all 𝑧𝑧 processes of the

database of dimension 𝑚𝑚 × 𝑧𝑧, respectively; r and �̃�𝑟 are vectors of environmental

burdens and requirements for sectors or individual processes of dimension 𝑚𝑚 ×

1, respectively; and

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δ = the direct requirement of the sector S or the process U.

B.3.3 Matrix Augmentation

The Matrix Augmentation method uses equations that are closely related to a pure

input-output analysis (see equation (B.1)). The difference lies in the dimension of each

element of the equation. Here, 𝑥𝑥 sectors are added to the 𝐴𝐴 and 𝑅𝑅 matrices, by

disaggregating an existing sector or adding a hypothetical sector to the economy. The

dimensions of the original matrices are therefore modified, as shown in equation (B.13) and

(B.14).

𝐴𝐴′ = �

𝑎𝑎1,1 ⋯ 𝑎𝑎1,𝑛𝑛 𝑎𝑎1,𝑛𝑛+𝑥𝑥

⋮ ⋱ ⋮𝑎𝑎𝑛𝑛,1𝑎𝑎𝑛𝑛+𝑥𝑥,1

⋯𝑎𝑎𝑛𝑛,𝑛𝑛 𝑎𝑎𝑛𝑛,𝑛𝑛+𝑥𝑥𝑎𝑎𝑛𝑛+𝑥𝑥,𝑛𝑛 𝑎𝑎𝑛𝑛+𝑥𝑥,𝑛𝑛+𝑥𝑥

� (B.13)

𝑅𝑅′ = �𝑟𝑟1,1 ⋯ 𝑟𝑟1,𝑛𝑛 𝑟𝑟1,𝑛𝑛+𝑥𝑥

⋮ ⋱ ⋮𝑟𝑟𝑚𝑚,1 ⋯ 𝑟𝑟𝑚𝑚,𝑛𝑛 𝑟𝑟𝑚𝑚,𝑛𝑛+𝑥𝑥

� (B.14)

The mathematical form of the Matrix Augmentation method (see Figure 3 (c) and

(d)) can then be summarised as per equation (B.15).

𝑞𝑞h = 𝑅𝑅′(𝐼𝐼 − 𝐴𝐴′)−1𝑦𝑦′h (B.15)

Where:

𝑞𝑞 = the vector of total environmental burdens and requirements of the sector 𝑆𝑆

associated with the hybrid component 𝐻𝐻, of dimension 𝑚𝑚 × 1;

𝐴𝐴′ = the augmented input-output matrix, including 𝑥𝑥 additional sectors, of

dimension (𝑛𝑛 + 𝑥𝑥) × (𝑛𝑛 + 𝑥𝑥);

55

𝑅𝑅′ = the augmented matrix of 𝑚𝑚 environmental burdens and requirements

associated with all 𝑛𝑛 + 𝑥𝑥 sectors of the economy, of dimension 𝑚𝑚 × (𝑛𝑛 + 𝑥𝑥);

𝐼𝐼 = the identity matrix of dimension (𝑛𝑛 + 𝑥𝑥) × (𝑛𝑛 + 𝑥𝑥); and

𝑦𝑦′ = the vector of final demand for the sector 𝑆𝑆 associated with the hybrid

component 𝐻𝐻, of dimension (𝑛𝑛 + 𝑥𝑥) × 1.

B.3.4 Integrated

The mathematical form of the Integrated method (see Figure 3 (e)) is generally

summarised as per equation (B.16) below. It differs from other methods in that the

relationship between the input-output and process components are solved internally within

the integrated matrix �̅�𝐴, using 𝐶𝐶𝑢𝑢 and 𝐶𝐶𝑑𝑑 the upstream and downstream cut-off matrices,

respectively. The upstream cut-off matrix represents inputs needed from processes within

the technological matrix, but for which no process data is available. This enables pulling

information from the input-output matrix to fill these gaps. On the other hand, the

downstream cut-off matrix represents the way in which products and services modelled in

the technological matrix are distributed to other parts of the economy.

𝑞𝑞h = �𝑅𝑅� 00 𝑅𝑅

� � 𝑇𝑇 −𝐶𝐶d−𝐶𝐶u 𝐼𝐼 − 𝐴𝐴

�−1�𝑦𝑦�u0 �

(B.16)

Equation (B.16) can then be further simplified as equation (B.17) below.

𝑞𝑞h = 𝑅𝑅��̅�𝐴−1𝑦𝑦�h (B.17)

Where:

56

𝑞𝑞 = the vector of total environmental burdens and requirements of the hybrid

component 𝐻𝐻 being analysed, of dimension 𝑚𝑚 × 1;

𝑇𝑇 = the technological matrix of dimension 𝑧𝑧 × 𝑧𝑧;

𝐴𝐴 = the input-output matrix of dimension 𝑛𝑛 × 𝑛𝑛;

𝑦𝑦� = the vector of final demand for the process 𝑈𝑈 associated with the hybrid

component 𝐻𝐻, of dimension 𝑧𝑧 × 1, respectively; and 𝑦𝑦� the hybrid vector of final

demand;

𝐼𝐼 = the identity matrix of dimension 𝑛𝑛 × 𝑛𝑛;

𝐶𝐶𝑢𝑢 = the upstream cut-off matrix of dimension 𝑛𝑛 × 𝑧𝑧;

𝐶𝐶𝑑𝑑 = the downstream cut-off matrix of dimension 𝑧𝑧 × 𝑛𝑛;

𝑅𝑅 and 𝑅𝑅� = the matrices of 𝑚𝑚 environmental burdens and requirements for all 𝑛𝑛

sectors of the economy and of dimension 𝑚𝑚 × 𝑛𝑛 , and all 𝑧𝑧 processes of the

database of dimension 𝑚𝑚 × 𝑧𝑧, respectively; and 𝑅𝑅� the sum of 𝑅𝑅 and 𝑅𝑅�; and

�̅�𝐴 = the integrated matrix of dimension (𝑛𝑛 + 𝑧𝑧) × (𝑛𝑛 + 𝑧𝑧).